The anomalous transmission properties of zero-permittivity ultranarrow channels are used to boost Kerr nonlinearities and achieve switching and bistable response for moderate optical intensities. Strong field enhancement, uniform all along the channel, is a typical feature of ε-near-zero supercoupling and is shown to be particularly suited to enhance nonlinear effects. This is obtained by designing narrow apertures at cutoff in a plasmonic screen. We show that this nonlinear mechanism can significantly outperform nonlinearities in traditional Fabry-Pérot resonant gratings.